Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $z \neq 0$. $t = \dfrac{-8}{27z + 9} \div \dfrac{8z}{z(3z + 1)} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $t = \dfrac{-8}{27z + 9} \times \dfrac{z(3z + 1)}{8z} $ When multiplying fractions, we multiply the numerators and the denominators. $t = \dfrac{ -8 \times z(3z + 1) } { (27z + 9) \times 8z } $ $ t = \dfrac {-8 \times z(3z + 1)} {8z \times 9(3z + 1)} $ $ t = \dfrac{-8z(3z + 1)}{72z(3z + 1)} $ We can cancel the $3z + 1$ so long as $3z + 1 \neq 0$ Therefore $z \neq -\dfrac{1}{3}$ $t = \dfrac{-8z \cancel{(3z + 1})}{72z \cancel{(3z + 1)}} = -\dfrac{8z}{72z} = -\dfrac{1}{9} $